A history of Greek mathematics - Wilbourhall.org

396 PAPPUS OF ALEXANDRIA
synthetical deduction, for which he claims that it is clearer
and shorter. We have first propositions (with auxiliary
lemmas) about the perpendiculars from the centre of the
circumscribing sphere to a face of (a) the octahedron, (b) the
icosahedron (Props. 39, 43), then the proposition that, if a
dodecahedron and an icosahedron be inscribed in the same
sphere, the same small circle in the sphere circumscribes both
the pentagon of the dodecahedron and the triangle of the
icosahedron (Prop. 48) ; this last is the proposition proved by
Hypsicles in the so-called Book XIV of Euclid ' ', Prop. 2, and
Pappus gives two methods of proof, the second of which (chap.
56) corresponds to that of Hypsicles. Prop. 49 proves that
twelve of the regular pentagons inscribed in a circle are together
greater than twenty of the equilateral triangles inscribed in
the same circle. The final propositions proving that the cube
is greater than the pyramid with the same surface, the octahedron
greater than the cube, and so on, are Props. 52-6
(chaps. 60-4), Of Pappus's auxiliary propositions, Prop. 41
is practically contained in Hypsicles's Prop. 1, and Prop. 44
in Hypsicles's last lemma; but otherwise the exposition is
different.
Book VI.
On the contents of Book VI we can be brief.
It is mainly
astronomical, dealing with the treatises included in the socalled
Little Astronomy, that is, the smaller astronomical
treatises which were studied as an introduction to the great
Syntaxia of Ptolemy. The preface says that many of those
who taught the Treasury of Astronomy, through a careless
understanding of the propositions, added some things as being
necessary and omitted others as unnecessary. Pappus mentions
at this point an incorrect addition to Theodosius, Sphaerica,
III. 6, an omission from Euclid's Phaenomena, Prop. 2, an
inaccurate representation of Theodosius, On Days and Nights,
Prop. 4, and the omission later of certain other things as
being unnecessary. His object is to put these mistakes
right. Allusions are also found in the Book to Menelaus's
Sphaerica, e.g. the statement (p. 476. 16) that Menelaus in
his Sphaerica called a spherical triangle TpLnXtvpov, three-side.